Problem: A circle has a radius of $9$. An arc in this circle has a central angle of $310^\circ$. What is the length of the arc? ${18\pi}$ ${310^\circ}$ $\color{#DF0030}{\dfrac{31}{2}\pi}$ ${9}$
Solution: First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (9) = 18\pi$ The ratio between the arc's central angle $\theta$ and $360^\circ$ is equal to the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{360^\circ} = \dfrac{s}{c}$ $\dfrac{310^\circ}{360^\circ} = \dfrac{s}{18\pi}$ $\dfrac{31}{36} = \dfrac{s}{18\pi}$ $\dfrac{31}{36} \times 18\pi = s$ $\dfrac{31}{2}\pi = s$